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Tangent to a circle
Geometry construction using a compass and straightedge
Click on 'Next' to go through the construction one step at a time, or click on 'Run' to let it run without stopping.
(If there is no image below, see support page.)
| After doing this |
Your work should look like this |
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We start with a point P somewhere on a given circle, with center point O.
If the center is not given, you can use: "Finding the center of a circle with compass and straightedge or ruler",
or
"Finding the center of a circle with any right-angled object".
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| 1. Draw a straight line through the center O of the circle and the point P right across the circle.
This is a diameter of the circle. |
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| 2. Mark a point Q anywhere. For best accuracy, avoid putting it too close to the diameter line. |
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| 3. Place the compass on the point Q just drawn, and set it's width to the point P. |
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| 4. Without changing the compass width, draw an arc across the diameter line, creating point R. |
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| 5. Again, without changing the compass width, draw another arc on the opposite side of Q. |
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| 6. Using the straightedge, draw a line through R and Q, extending it onwards so it crosses the arc just drawn. Mark this point S. |
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| 7. Using the straightedge, draw a line through P and S, extending it in both directions. |
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| 8. Done. The line just drawn is the tangent to the circle O through point P. |
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Proof
Recall that the tangent to a circle is perpendicular to the radius at the contact point. (See Tangent definition page).
This construction uses that fact in reverse. By constructing a line at right angles to the diameter it must be the tangent line.
The method used to construct the perpendicular is exactly the same as the one used in
"Construct a perpendicular at the end of a ray".
The image below is the final drawing above with the red lines added.
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Argument |
Reason |
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Line segments QS, QR are congruent to QP |
QS and QR were both drawn with the compass width set to QP |
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The points R, S, P lie on a circle, center Q (shown in red above). |
QR, QS, QP are radii of the red circle. |
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RS is a diameter of the circle center Q |
It is straight because it was drawn with a straightedge, and passes through the center.
See Diameter of a circle |
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Angle RPS is a right angle |
The angle inscribed in a semicircle is always a right angle (90°).
See Angle inscribed in a semicircle. |
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OP is a radius of the given circle |
A line from the center to a point on the circle.
See Radius of a circle. |
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PS is a tangent to the given circle
Q.E.D. |
It touches the circle at one point, and is perpendicular to a radius at that point.
See Tangent to a circle |
Try it yourself
Click here for a printable tangents construction worksheet with some problems to try.
When you get to the page, use the browser print command to print as many as you wish. The printed output is not copyright.
Constructions pages on this site
Lines
Angles
Triangles
Triangle Centers
Circles, Arcs and Ellipses
Non-Euclidean constructions
(C) 2009 Copyright John Page
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